// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2015 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_LEAST_SQUARE_CONJUGATE_GRADIENT_H
#define EIGEN_LEAST_SQUARE_CONJUGATE_GRADIENT_H

namespace Eigen {

namespace internal {

/** \internal Low-level conjugate gradient algorithm for least-square problems
 * \param mat The matrix A
 * \param rhs The right hand side vector b
 * \param x On input and initial solution, on output the computed solution.
 * \param precond A preconditioner being able to efficiently solve for an
 *                approximation of A'Ax=b (regardless of b)
 * \param iters On input the max number of iteration, on output the number of performed iterations.
 * \param tol_error On input the tolerance error, on output an estimation of the relative error.
 */
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
EIGEN_DONT_INLINE void
least_square_conjugate_gradient(const MatrixType& mat,
								const Rhs& rhs,
								Dest& x,
								const Preconditioner& precond,
								Index& iters,
								typename Dest::RealScalar& tol_error)
{
	using std::abs;
	using std::sqrt;
	typedef typename Dest::RealScalar RealScalar;
	typedef typename Dest::Scalar Scalar;
	typedef Matrix<Scalar, Dynamic, 1> VectorType;

	RealScalar tol = tol_error;
	Index maxIters = iters;

	Index m = mat.rows(), n = mat.cols();

	VectorType residual = rhs - mat * x;
	VectorType normal_residual = mat.adjoint() * residual;

	RealScalar rhsNorm2 = (mat.adjoint() * rhs).squaredNorm();
	if (rhsNorm2 == 0) {
		x.setZero();
		iters = 0;
		tol_error = 0;
		return;
	}
	RealScalar threshold = tol * tol * rhsNorm2;
	RealScalar residualNorm2 = normal_residual.squaredNorm();
	if (residualNorm2 < threshold) {
		iters = 0;
		tol_error = sqrt(residualNorm2 / rhsNorm2);
		return;
	}

	VectorType p(n);
	p = precond.solve(normal_residual); // initial search direction

	VectorType z(n), tmp(m);
	RealScalar absNew = numext::real(normal_residual.dot(p)); // the square of the absolute value of r scaled by invM
	Index i = 0;
	while (i < maxIters) {
		tmp.noalias() = mat * p;

		Scalar alpha = absNew / tmp.squaredNorm();	// the amount we travel on dir
		x += alpha * p;								// update solution
		residual -= alpha * tmp;					// update residual
		normal_residual = mat.adjoint() * residual; // update residual of the normal equation

		residualNorm2 = normal_residual.squaredNorm();
		if (residualNorm2 < threshold)
			break;

		z = precond.solve(normal_residual); // approximately solve for "A'A z = normal_residual"

		RealScalar absOld = absNew;
		absNew = numext::real(normal_residual.dot(z)); // update the absolute value of r
		RealScalar beta = absNew / absOld; // calculate the Gram-Schmidt value used to create the new search direction
		p = z + beta * p;				   // update search direction
		i++;
	}
	tol_error = sqrt(residualNorm2 / rhsNorm2);
	iters = i;
}

}

template<typename _MatrixType,
		 typename _Preconditioner = LeastSquareDiagonalPreconditioner<typename _MatrixType::Scalar>>
class LeastSquaresConjugateGradient;

namespace internal {

template<typename _MatrixType, typename _Preconditioner>
struct traits<LeastSquaresConjugateGradient<_MatrixType, _Preconditioner>>
{
	typedef _MatrixType MatrixType;
	typedef _Preconditioner Preconditioner;
};

}

/** \ingroup IterativeLinearSolvers_Module
  * \brief A conjugate gradient solver for sparse (or dense) least-square problems
  *
  * This class allows to solve for A x = b linear problems using an iterative conjugate gradient algorithm.
  * The matrix A can be non symmetric and rectangular, but the matrix A' A should be positive-definite to guaranty
  stability.
  * Otherwise, the SparseLU or SparseQR classes might be preferable.
  * The matrix A and the vectors x and b can be either dense or sparse.
  *
  * \tparam _MatrixType the type of the matrix A, can be a dense or a sparse matrix.
  * \tparam _Preconditioner the type of the preconditioner. Default is LeastSquareDiagonalPreconditioner
  *
  * \implsparsesolverconcept
  *
  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
  * and NumTraits<Scalar>::epsilon() for the tolerance.
  *
  * This class can be used as the direct solver classes. Here is a typical usage example:
	\code
	int m=1000000, n = 10000;
	VectorXd x(n), b(m);
	SparseMatrix<double> A(m,n);
	// fill A and b
	LeastSquaresConjugateGradient<SparseMatrix<double> > lscg;
	lscg.compute(A);
	x = lscg.solve(b);
	std::cout << "#iterations:     " << lscg.iterations() << std::endl;
	std::cout << "estimated error: " << lscg.error()      << std::endl;
	// update b, and solve again
	x = lscg.solve(b);
	\endcode
  *
  * By default the iterations start with x=0 as an initial guess of the solution.
  * One can control the start using the solveWithGuess() method.
  *
  * \sa class ConjugateGradient, SparseLU, SparseQR
  */
template<typename _MatrixType, typename _Preconditioner>
class LeastSquaresConjugateGradient
	: public IterativeSolverBase<LeastSquaresConjugateGradient<_MatrixType, _Preconditioner>>
{
	typedef IterativeSolverBase<LeastSquaresConjugateGradient> Base;
	using Base::m_error;
	using Base::m_info;
	using Base::m_isInitialized;
	using Base::m_iterations;
	using Base::matrix;

  public:
	typedef _MatrixType MatrixType;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	typedef _Preconditioner Preconditioner;

  public:
	/** Default constructor. */
	LeastSquaresConjugateGradient()
		: Base()
	{
	}

	/** Initialize the solver with matrix \a A for further \c Ax=b solving.
	 *
	 * This constructor is a shortcut for the default constructor followed
	 * by a call to compute().
	 *
	 * \warning this class stores a reference to the matrix A as well as some
	 * precomputed values that depend on it. Therefore, if \a A is changed
	 * this class becomes invalid. Call compute() to update it with the new
	 * matrix A, or modify a copy of A.
	 */
	template<typename MatrixDerived>
	explicit LeastSquaresConjugateGradient(const EigenBase<MatrixDerived>& A)
		: Base(A.derived())
	{
	}

	~LeastSquaresConjugateGradient() {}

	/** \internal */
	template<typename Rhs, typename Dest>
	void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
	{
		m_iterations = Base::maxIterations();
		m_error = Base::m_tolerance;

		internal::least_square_conjugate_gradient(matrix(), b, x, Base::m_preconditioner, m_iterations, m_error);
		m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
	}
};

} // end namespace Eigen

#endif // EIGEN_LEAST_SQUARE_CONJUGATE_GRADIENT_H
